{"id":"32911975-dfcd-4b0b-aac2-c7e0c493f21d","arxiv_id":"1908.07092","paper_version":11,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The leading eigenvalue of infinitely large random directed graphs is finite and universal, governed by c(ρ+1) and the coupling moments, so large directed dynamical systems can be stable.","lead":"This paper derives exact formulas for the leading eigenvalue of random directed graphs with prescribed degree distributions, showing that the stability of large dynamical systems on such graphs depends only on mean degree, degree correlations, and coupling statistics. The result implies that directed networks can remain stable at arbitrarily large size, unlike their undirected counterparts.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The derivation of the central formulas (53) and (57) rests on an unproven 'stable eigenvalue' assumption that is not literally satisfied by finite matrices for boundary eigenvalues; a direct deletion test would settle whether the assumption holds asymptotically.","rationale":"The paper's central contribution is the exact-looking formula λ* = max{-d+c(ρ+1)⟨J⟩, -d+√(c(ρ+1)⟨J²⟩)} and the resulting universal phase diagram. All of the analytical content, including the definition of λ_b, flows from the recursive distributional equations (48)-(49). Those equations are closed by asserting that the eigenvector entries of the cavity matrix A^(j) coincide with those of A for the same eigenvalue λ (Eq. E3). This assertion is derived in Appendix F by assuming λ is an eigenvalue common to A and all A^(j). For an isolated outlier this is plausible in the limit, but for a boundary point of a continuous spectrum it is not literally true, and the paper provides no limiting argument. The authors themselves state in Sec. VII that the results are conjectures and cite a proof only for the leading eigenvalue of directed Erdős-Rényi graphs with J=1. The concern is therefore not about a minor step: it is the hinge on which the universality claim turns. The suggested deletion experiment directly probes whether the stability condition holds asymptotically; if it does, the conditional acceptance is justified, and if it does not, the central formula needs revision. I agree with the reader's identification of this as the weakest point and therefore do not change the CONDITIONAL verdict.","tokens_in":49009,"tokens_out":19635,"duration_ms":203104,"concrete_test":"Perform a deletion test in the gapless regime: for directed Erdős-Rényi graphs with c=2, Gaussian weights with zero mean and unit variance, and d=0, compute the eigenvalue of A_n closest to the predicted boundary |λ_b|=√2 for n=10^3, 10^4, and 10^5. Delete a uniformly random node j, compute the eigenvalue of A_n^(j) closest to the same value, and measure the mean absolute difference δ_n over many realizations. If δ_n does not tend to zero as n grows, the stable-eigenvalue assumption behind Eqs. (48)-(49) fails for boundary eigenvalues, and Eq. (53) is not justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The recursion relations (48)-(49) are the sole route to the boundary formula |λ_b+d|² = c(ρ+1)⟨J²⟩ and the outlier formula λ_isol = -d+c(ρ+1)⟨J⟩. Appendix F derives these recursions by assuming that the eigenvalue λ of interest is stable, i.e., that λ is an eigenvalue of A and of every principal submatrix A^(j) (paragraph before Eq. F2 and Eq. F2). For λ_b, a boundary point of the limiting continuous spectrum, this condition fails for finite matrices: λ_b is a limit point of the spectrum, not an eigenvalue of A, so the Schur-based residue formula (F2) and the equality R_k = R_k^(j) (E3) are only formal. The paper acknowledges in Sec. VII that the general results are conjectures and that a proof exists only for directed Erdős-Rényi graphs with J=1 (Ref. [97]). Since the universality claim depends entirely on these recursions, the central result is not established without the stability condition. The finite-size diagonalizations at n=4000 corroborate the final formulas but do not test the stability assumption itself.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the leading eigenvalue and associated right/left eigenvectors of sparse non-Hermitian random matrices of the form A = -d I + J∘C, where C is the adjacency matrix of a directed configuration-model graph with prescribed joint in/out degree distribution and J has i.i.d. entries. Using recursions derived from the Schur formula under a locally tree-like and oriented assumption, the authors obtain closed-form expressions for the boundary of the continuous spectrum, |λ_b+d|² = c(ρ+1)⟨J²⟩, and for the deterministic outlier λ_isol = -d + c(ρ+1)⟨J⟩. From these they construct a universal stability phase diagram depending only on c(ρ+1), ⟨J⟩, ⟨J²⟩, and d, and they characterize the destabilizing mode as ferromagnetic or spin-glass-like. The analytical predictions are compared with full diagonalization for n=4000 on Poisson, geometric, and power-law directed random graphs, with good agreement outside the most strongly fluctuating regimes. The paper also extends the formalism to diagonal disorder and to undirected random graphs and discusses the role of finite oriented cycles, which can create stochastic outliers. The general results are presented as conjectures, with a rigorous proof available only for directed Erdős-Rényi graphs with J=1 via Ref. [97].","tokens_in":49291,"tokens_out":7199,"duration_ms":78416,"significance":"If the central formulas are correct, the paper delivers a strikingly simple and universal criterion for the stability of large dynamical systems on directed random graphs, and it establishes that such systems can remain stable at infinite size even for degree distributions with unbounded support, in contrast to their undirected counterparts. The paper's strengths include a self-contained re-derivation of the cavity recursion relations through the Schur formula, a clear identification of the conjecture status in Sec. VII, and extensive finite-size numerical checks with no parameter fitting: the theoretical curves use only the prescribed ensemble parameters c, ρ, ⟨J⟩, and ⟨J²⟩. The results, if established, would have broad relevance for ecological, neural, and financial network stability, and the universal phase diagram is a useful organizing principle for sparse non-Hermitian random matrix theory.","major_comments":[{"comment":"The derivation of the central recursions (48)-(49), and hence of the boundary formula (53) and the outlier formula (57), assumes that the eigenvalue λ is 'stable', i.e., that λ is an eigenvalue of A and also of every single-node-deleted principal submatrix A^(j). For the boundary eigenvalue λ_b of ∂σ_ac this assumption is not literally satisfied for finite matrices: λ_b is a limit point of the spectrum rather than an isolated eigenvalue, so the residue identity (F2) and the equality R_k = R_k^(j) in (E3) are only formal. The finite-size diagonalizations in Sec. V corroborate the final formulas but do not test deletion stability of λ_b itself. Since Eq. (53) is load-bearing for the universal phase diagram, the paper should either provide a rigorous asymptotic justification (for example, a regularization that takes η→0 after n→∞, or a local weak convergence proof) or add a direct numerical deletion test and reframe the general claims as well-supported conjectures rather than exact theory.","section":"Appendix F"},{"comment":"The numerical evidence for the headline claim that systems remain stable for degree distributions with unbounded support is weakest precisely in the regime of unbounded second moments. For the uncorrelated power-law ensemble of Eq. (99), deviations from Eqs. (57) and (53) are substantial for a≲3, and for the perfectly correlated ensemble of Eq. (100) for a≲4; these deviations are attributed to finite-size effects, but only n=2000 and n=4000 are shown, with no finite-size scaling or extrapolation to n→∞. A finite-size scaling analysis is needed to support the claim that the deviations vanish in the infinite-size limit and that the unbounded-support stability result is numerically established.","section":"Sec. V C"}],"minor_comments":[{"comment":"In the paragraph after Eq. (I2), the equation numbers appear to be swapped: solving Eq. (I1) for ⟨R⟩_q≠0 yields Eq. (108) for the outlier, while setting ⟨R⟩_q=0 in Eq. (I2) yields Eq. (109) for the boundary, not the reverse as written.","section":"Appendix I"},{"comment":"The axis labels in Figs. 3 and 4 render the effective connectivity as 'c( + 1)', with the ρ missing; the labels should read c(ρ+1).","section":"Figs. 3 and 4"},{"comment":"The abstract and Sec. III describe the theory as 'exact', while Sec. VII states that Eqs. (53), (57), (58), (48), (49), (45), (56), and (60) are conjectures for all ensembles except directed Erdős-Rényi graphs with J=1. The wording should be aligned so that the conjecture status is transparent already in the abstract and introduction.","section":"Abstract and Sec. VII"},{"comment":"There is a typo in the sentence 'setting ⟨R⟩_q = 0 in Eq. (I2), ew obtain Eq. (108)' where 'ew' should read 'we'; the same sentence also contains the equation-number swap noted above.","section":"Appendix I"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about the conjecture status of its main results, and the numerical evidence is strong in the Poisson and geometric ensembles. The main risk is the gap between the 'exact theory' framing and the formal nature of the stable-eigenvalue assumption for λ_b. If the journal accepts well-supported conjectures in this area, the requested deletion test, finite-size scaling, and reframing should be sufficient; if the journal requires proofs for load-bearing statements, the gap would be grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is the strongest version so far of the cavity-method results for the leading edge of sparse directed random graphs. It goes from the uncorrelated-degree case in the authors' earlier PRL to arbitrary prescribed indegree/outdegree distributions, adds degree correlations ρ, eigenvector statistics, and a stability phase diagram, and it supports the formulas with systematic diagonalizations at n = 4000, including power-law tails. The central claim—that λ* = max{λ_isol, |λ_b + d| − d} with λ_isol = −d + c(ρ+1)⟨J⟩ and |λ_b + d|² = c(ρ+1)⟨J²⟩—is clean and, if right, resolves a question many people have circled around: why directed networks can remain stable at large size while undirected ones cannot. The Schur-formula derivation in Appendices E–G is a real pedagogical improvement over importing the recursions, and the paper is honest that the general statements are conjectures (Sec. VII), with a proof available only for directed Erdős–Rényi graphs with J = 1.\n\nThe soft spot is exactly the one the authors flag. The recursion relations (48)–(49), and hence the formulas (53) and (57), rely on λ being a stable eigenvalue: an eigenvalue of A and of every principal submatrix A^(j) (Appendix F). For λ_b on the boundary of the continuous spectrum this is not literally true for finite matrices—λ_b is a limit point, not an eigenvalue—so the residue formula and the equality R_k = R_k^(j) are formal. The finite-n diagonalizations at n = 4000 are impressive in scope, but they test the final formulas, not the stability assumption itself. A direct deletion test—compute the spectrum of A^(j) for large n and check that the relevant eigenvalue persists—would settle whether the assumption holds asymptotically. I do not think this kills the paper. The conjecture is well-defined, the numerics are broad, and the physics conclusion (stability controlled by c(ρ+1), ⟨J⟩, ⟨J²⟩, and d) is exactly what one would want. But the abstract's \"exact theory\" overstates what is actually proven; the Discussion's \"conjectures\" is the accurate framing, and the authors should align the two before publication.\n\nMinor: no code or data archive is provided, which makes the population-dynamics solution of (48)–(49) in Fig. 8 hard to reproduce; the algorithm is described, but raw data would help. Self-citations are appropriate here—the earlier PRL is the direct ancestor and the recursions are re-derived, not imported.\n\nWho should read it: anyone working on sparse non-Hermitian random matrix theory, ecological or neural stability, or spectral algorithms for directed networks. It deserves a serious referee: referee time should go to checking the stability assumption and the moment-closure steps in Appendix G, not to desk-rejecting. I would accept it for review.","headline":"A substantial, numerically well-supported extension of the cavity approach to directed sparse graphs; the main formulas are honestly labeled conjectures, and the load-bearing stable-eigenvalue assumption is formal for boundary eigenvalues and should be tested directly.","tokens_in":49744,"tokens_out":2516,"would_cite":true,"duration_ms":26968,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C80","15B52","60B20","34D20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For infinitely large random directed graphs, stability reduces to a four-parameter leading-eigenvalue formula.","keywords":["linear stability","random directed graphs","leading eigenvalue","degree correlations","non-Hermitian random matrices","cavity method","configuration model","spectral outliers"],"falsifier":"Fix unweighted couplings $J_{jk}=1$ on directed configuration-model graphs with $c(\\rho+1)>1$ and growing size $n$: the paper predicts $\\lambda_1 \\to -d+c(\\rho+1)$ with vanishing variance. Observing $\\lambda_1$ drifting with $n$, diverging, or fluctuating with non-vanishing sample variance would falsify the central formula. Repeating the check on power-law degree distributions with exponent just above $2$ tests the unbounded-support stability claim directly.","tokens_in":48806,"feed_emoji":"🕸️","tokens_out":11084,"duration_ms":93532,"temperature":0.7,"pith_summary":"This paper asks when a fixed point of a large dynamical system coupled through a random directed graph remains stable against small perturbations. The authors claim that in the infinite-size limit the typical leading eigenvalue of the Jacobian, whose real part decides stability, is given by a closed formula involving only the effective mean degree $c(\\rho+1)$, the first two moments of the coupling distribution, and the relaxation rate $d$. From this formula they derive a universal phase diagram: the stability boundary, the spectral gap, and the nature of the destabilizing mode all depend on the same few parameters. They also argue for a qualitative contrast with undirected networks, where the leading eigenvalue grows as $\\sqrt{k_{\\max}}$: on locally tree-like directed graphs the leading eigenvalue stays finite, so infinitely large systems can be stable even when the degree distribution has unbounded support. The general results are stated as conjectures, corroborated by numerical diagonalization of finite random graphs and by rigorous results in a special Poisson-degree case with unit weights.","feed_headline":"Directed random networks can stay stable at any size","feed_subtitle":"The typical largest eigenvalue depends only on mean degree, degree correlation, coupling moments, and decay rate.","key_machinery":"The machinery is the locally tree-like and oriented property of random directed configuration-model graphs: with probability one, the finite neighbourhood of any node is an oriented tree, so there are no finite feedback loops that amplify perturbations. On such graphs the Schur-complement recursion for eigenvector entries closes, producing the distributional equations (48) and (49) for the entry distribution $p_R$ and the associated edge-rooted distribution $q_R$. Eigenvalues of interest are located by asking for which $\\lambda$ these equations admit a normalizable solution: the outlier $\\lambda_{\\mathrm{isol}}$ corresponds to a solution with nonzero first moment, while the continuum boundary $\\lambda_{\\mathrm{b}}$ corresponds to a zero-mean solution. Finite-length oriented rings, whose eigenvalues lie on a circle of radius $\\gamma = (\\prod_j |J_j|)^{1/\\ell}$ centred at $-d$, supply the residual stochastic outliers that give $\\lambda_1$ a nonzero variance.","core_discovery":"The paper's central discovery is that for an infinitely large directed configuration-model graph with degree distribution $p_{K^{\\mathrm{in}},K^{\\mathrm{out}}}$ and i.i.d. couplings $J_{jk}$, the typical leading eigenvalue is $\\lambda^{*} = \\max\\{\\lambda_{\\mathrm{isol}}, |\\lambda_{\\mathrm{b}}+d|-d\\}$, with $\\lambda_{\\mathrm{isol}} = -d + c(\\rho+1)\\langle J\\rangle$ and $|\\lambda_{\\mathrm{b}}+d|^{2} = c(\\rho+1)\\langle J^{2}\\rangle$, whenever the graph has a giant strongly connected component, $c(\\rho+1)>1$; below that threshold the spectrum collapses to $\\{-d\\}$ apart from stochastic outliers created by oriented rings. The leading eigenvalue is thus finite and self-averaging in the giant-component regime, in contrast to undirected sparse graphs. The same recursive scheme yields the statistics of the eigenvector entries and identifies the destabilizing mode: when the outlier $\\lambda_{\\mathrm{isol}}$ dominates, the mode is ferromagnetic with positive mean entry; when the continuum boundary $\\lambda_{\\mathrm{b}}$ dominates, the mode is spin-glass-like with zero mean entry.","pith_inferences":["Beyond the paper, if typical stability depends only on $c(\\rho+1)$, $\\langle J\\rangle$, and $\\langle J^2\\rangle$, then any rewiring that preserves these three parameters cannot change typical linear stability; meaningful interventions must act on at least one of them.","Beyond the paper, the boundary-and-outlier logic should carry over to discrete-time dynamics, where stability is governed by the spectral radius rather than the largest real part; making this explicit would require the distribution of complex outliers from oriented rings, not just the real edge.","Beyond the paper, a direct test of the conjecture is deletion-stability: on finite power-law graphs, delete single nodes and check whether the candidate boundary eigenvalue persists; a failure in the heavy-tailed regime would mark exactly where the unbounded-support claim breaks down."],"forward_implications":["With a giant strongly connected component, the typical leading eigenvalue is deterministic: sample-to-sample fluctuations vanish as $n\\to\\infty$, and only rare oriented rings with large enough weights can produce outliers.","The phase boundary depends only on $c(\\rho+1)$, $\\langle J\\rangle$, $\\langle J^2\\rangle$, and $d$, so decreasing the indegree–outdegree correlation $\\rho$ stabilizes the system when coupling fluctuations are moderate, while increasing the mean coupling or its variance moves it toward instability.","For small coupling fluctuations the destabilizing mode is a ferromagnetic outlier whose position is independent of $\\langle J^2\\rangle$; for larger fluctuations the leading eigenvalue sits at the continuum boundary and the mode is spin-glass-like with zero mean.","Directed locally tree-like networks can remain stable at arbitrarily large size, including power-law degree distributions with unbounded support, because the leading eigenvalue does not diverge with the largest degree; this is the paper's main contrast with undirected networks.","In the dense limit the formula formally reduces to $n\\langle J\\rangle$ when $\\langle J\\rangle>0$ and $\\sqrt{n\\langle J^2\\rangle}$ otherwise, recovering the classical dense random-matrix results and showing the sparse theory contains the dense one as a limiting case."],"supporting_citations":[{"why":"Supplies the original cavity recursion for eigenvalue outliers in locally tree-like non-Hermitian matrices that this paper re-derives from the Schur formula.","marker":"[44]"},{"why":"Provides the spectral theory of sparse non-Hermitian random matrices used for the bulk spectrum and the spherical-model interpretation.","marker":"[45]"},{"why":"Gives the cavity approach to the spectral density of non-Hermitian sparse matrices that underlies the resolvent recursions.","marker":"[42]"},{"why":"Establishes the resolvent and local weak convergence framework invoked for the Schur-complement derivation.","marker":"[56]"},{"why":"Supplies the only known rigorous proof of the leading and subleading eigenvalue formulas in the unit-weight directed Poisson-degree case.","marker":"[97]"},{"why":"The classical dense-system instability benchmark that the paper contrasts with and recovers in the dense limit.","marker":"[20]"},{"why":"Reports the heuristic that Eq. (57) describes the largest eigenvalue of unweighted adjacency matrices, which the paper upgrades to an exact outlier expression.","marker":"[62]"}],"fun_headline_variants":["Stable at any size: directed graphs defy undirected instability","Directed networks: infinite size, finite stability","Size no obstacle: directed random graphs stay stable","Universal stability threshold in directed random graphs","Directed graphs: stability that scales to infinity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the boundary eigenvalue and the outlier eigenvalue remain eigenvalues after any single node is deleted; this 'stability of the eigenvalue' is assumed rather than proved, and the paper explicitly presents the general results as conjectures.","fun_headline_variants_meta":{"raw":{"variants":["Stable at any size: directed graphs defy undirected instability","Directed networks: infinite size, finite stability","Size no obstacle: directed random graphs stay stable","Universal stability threshold in directed random graphs","Directed graphs: stability that scales to infinity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000263,"raw_usage":{"total_tokens":1635,"prompt_tokens":1014,"completion_tokens":621,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":630,"completion_tokens_details":{"reasoning_tokens":551}},"tokens_in":630,"tokens_out":621,"duration_ms":6201,"temperature":1.0,"reasoning_tokens":551,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:26:10.646496+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix unweighted couplings $J_{jk}=1$ on directed configuration-model graphs with $c(\\rho+1)>1$ and growing size $n$: the paper predicts $\\lambda_1 \\to -d+c(\\rho+1)$ with vanishing variance. Observing $\\lambda_1$ drifting with $n$, diverging, or fluctuating with non-vanishing sample variance would falsify the central formula. Repeating the check on power-law degree distributions with exponent just above $2$ tests the unbounded-support stability claim directly.","supporting_citations":[{"cited_title":"Resolvent of large ran- dom graphs,","cited_arxiv_id":null,"evidence_quote":"Establishes the resolvent and local weak convergence framework invoked for the Schur-complement derivation."},{"cited_title":"Detection thresholds in very sparse matrix completion","cited_arxiv_id":"2005.06062","evidence_quote":"Supplies the only known rigorous proof of the leading and subleading eigenvalue formulas in the unit-weight directed Poisson-degree case."},{"cited_title":"Approximating the largest eigenvalue of network adjacency matrices,","cited_arxiv_id":null,"evidence_quote":"Reports the heuristic that Eq. (57) describes the largest eigenvalue of unweighted adjacency matrices, which the paper upgrades to an exact outlier expression."}],"review_version":1}